How do you extude along a specified axis?
Specifically when in edit mode and wanting to extrude along a specific axis orientation, you can set Transform Orientations to cursor. 
Then in the view panel, set the cursor rotation to the orientation you want to extrude along.

Then simply extrude. Pressing the key E starts an extrusion, then you press the axis key you want ie Z, you can use mouse to set visually, or  then just enter the number value.
Press Enterto commit the extrusion.
Alternately you can set Transform Orientations to normal, setting Transform Orientations to local will refer to the objects transform orientation.